The Efros–Shklovskii (ES) Coulomb gap in the one-electron density of localized states and the ES law of the variable range hopping conductivity were coined 50 years ago. The theory and its first confirmations were reviewed in the Shklovskii–Efros (SE) monograph published 40-years ago. This paper reviews the subsequent experimental evidence, theoretical advancements, and novel applications of the ES law. Out of hundreds of experimental validations of the ES law in a diverse range of materials, I focus on those where the dynamic range of conductivity exceeds four orders of magnitude. These include three- and two-dimensional semiconductors under both zero and high magnetic fields, localized phases in the quantum Hall effect, granular metals, nanocrystal arrays, and conducting polymers. Additionally, I discuss the non-ohmic ES law and the Coulomb gap near insulator–metal transition. Recent developments of other concepts of the SE book are also discussed.

1.
A. L.
Efros
and
B. I.
Shklovskii
, “
Coulomb gap and low temperature conductivity of disordered systems
,”
J. Phys. C
8
,
L49
(
1975
).
2.
A. L.
Efros
, “
Coulomb gap in disordered systems
,”
J. Phys. C
9
,
2021
(
1976
).
3.
B. I.
Shklovskii
and
A. L.
Efros
,
Electronic Properties of Doped Semiconductors
(
Springer-Verlag
,
Berlin
,
1984
).
4.
N.
Van Lien
,
Fiz. Tech. Poluprovodn.
18
,
335
(
1984
) [Sov. Phys. Semicond. 18, 207 (1984)].
5.
K. M.
Itoh
,
W. L.
Hansen
,
J. W.
Beeman
,
E. E.
Haller
,
J. W.
Farmer
, and
V. I.
Ozhogin
,
J. Low Temp. Phys.
93
,
307
(
1993
).
6.
W.
Schoepe
,
Z. Phys. B Condens. Matter
71
,
455
(
1988
).
7.
Dan
McCammon
,
Cryogenic Particle Detection
, (
Springer
,
Berlin, Heidelberg
,
2005
), arXiv:physics/0503086.
8.
A.
Consiglio
,
G.
Gatti
,
E.
Martino
,
L.
Moreschini
,
J. C.
Johannsen
,
K.
Prša
,
P. G.
Freeman
,
D.
Sheptyakov
,
H. M.
Rønnow
,
R.
Scopelliti
,
A.
Magrez
,
L.
Forró
,
C.
Schmitt
,
V.
Jovic
,
C.
Jozwiak
,
A.
Bostwick
,
E.
Rotenberg
,
T.
Hofmann
,
R.
Thomale
,
G.
Sangiovanni
,
D.
Di Sante
,
M.
Greiter
,
M.
Grioni
, and
S.
Moser
, “
Electron glass phase with resilient Zhang-Rice singlets in LiCuO3O3
,”
Phys. Rev. Lett.
132
,
126502
(
2024
).
9.
D. Yu, C.
Wang
,
B. L.
Wehrenberg
, and
P.
Guyot-Sionnest
,
Phys. Rev. Lett.
92
,
216802
1
(
2004
).
10.
B. L.
Greenberg
,
Z. L.
Robinson
,
K. V.
Reich
,
C.
Gorynski
,
B. N.
Voigt
,
L. F.
Francis
,
B. I.
Shklovskii
, and
E. S.
Aydil
, and
Uwe R.
Kortshagen
,
Nano Lett.
17
,
4634
(
2017
).
11.
H.
Zeng
,
C. T.
Black
,
R. L.
Sandstrom
,
P. M.
Rice
,
C. B.
Murray
, and
S.
Sun
, “
Magneto-transport of magnetite nanoparticle arrays
,”
Phys. Rev. B
73
,
020402
(
2006
).
12.
B.
Fisher
,
J.
Genossar
,
K. B.
Chashka
,
A.
Knizhnik
,
L.
Patlagan
, and
G. M.
Reisner
, “
Variable range hopping in A2MnReO6 (A = Ca, Sr, Ba)
,”
J. Appl. Phys.
104
,
033716
(
2008
).
13.
B.
Abeles
,
P.
Sheng
,
M. D.
Coutts
, and
Y.
Arie
, “
Structural and electrical properties of granular metal films
,”
Adv. Phys.
24
,
407
(
1975
).
14.
T.
Chui
,
G.
Deutscher
,
P.
Lindenfeld
, and
W. L.
McLean
, “
Conduction in granular aluminum near the metal-insulator transition
,”
Phys. Rev. B
23
,
6172
(
1981
).
15.
L.
Zuppiroli
,
M. N.
Bussac
,
S.
Paschen
,
O.
Chauvet
, and
L.
Forro
,
Phys. Rev. B
50
,
5196
(
1994
).
16.
F.
Zuo
,
M.
Angelopoulos
, and
A. G.
MacDiarmid
, and
Arthur J.
Epstein
,
Phys. Rev. B
36
,
3475
(
1987
).
17.
B.
Sixou
,
N.
Mermilliod
, and
J. P.
Travers
,
Phys. Rev. B
53
,
4509
(
1996
).
18.
C. O.
Yoon
,
M.
Reghu
,
D.
Moses
,
A. J.
Heeger
,
Y.
Cao
,
T.-A.
Chen
,
X.
Wu
, and
R. D.
Rieke
,
Synthetic Metals
75
,
229
(
1995
).
19.
Y.
Wang
,
S. S.
Mohite
,
L. B.
Bridwell
,
R. E.
Giedd
, and
C. J.
Sofield
,
J. Mater. Res.
8
,
388
(
1993
).
20.
21.
S.
Wang
,
M.
Ha
,
M.
Manno
,
C.
Daniel Frisbie
, and
C.
Leighton
,
Nature Commun.
3
,
1210
(
2013
).
22.
D.
Joung
and
S. I.
Khondaker
,
Phys. Rev. B
86
,
235423
(
2012
).
23.
R. M.
Rudenko
,
O. O.
Voitsihovska
,
A. A.
Abakumov
,
I. B.
Bychko
, and
V. N.
Poroshin
,
Diam. Relat. Mater.
141
,
110606
(
2024
).
24.
D. P.
Wang
,
D. E.
Feldman
,
B. R.
Perkins
,
A. J.
Yin
,
G. H.
Wang
,
J. M.
Xu
, and
A.
Zaslavsky
, arXiv:cond-mat/0610747.
25.
A. I.
Yakimov
,
A. V.
Dvurechenskii
,
V. V.
Kirienko
,
Yu. I.
Yakovlev
,
A. I.
Nikiforov
, and
C. J.
Adkins
,
Phys. Rev. B
61
,
10868
(
2000
).
26.
B. I.
Shklovskii
, “
Variable-range hopping conduction in a strong magnetic field
,”
Zh. Eksp. Theor. Fiz. Pis. Red.
36
,
43
(
1982
) [Sov. Phys.-JETP Lett. 36, 53 (1982)].
27.
H.
Tokumoto
,
R.
Mansfield
, and
M. J.
Lea
, “
Correlation effects in variable-range hopping in n-InSb
,”
Solid State Commun.
35
,
961
(
1980
).
28.
F. W.
Van Keuls
,
X. L.
Hu
,
H. W.
Jiang
, and
A. J.
Dahm
, “
Screening of the Coulomb interaction in two-dimensional variable-range hopping
,”
Phys. Rev. B
56
,
1161
(
1997
).
29.
G.
Ebert
,
K.
von Klitzing
,
C.
Probst
,
E.
Schuberth
,
K.
Ploog
, and
G.
Weinmann
, “
Two-dimensional magneto-quantum transport in GaAs-AlxGa1–xAs heterostructures under non-ohmic conditions
,”
Solid State Commun.
45
,
625
(
1983
).
30.
A.
Briggs
,
Y.
Gulder
,
J. P.
Vierne
,
M.
Voos
,
J. R.
Hirtz
, and
J. P.
Razeghi
, “
Low-temperature investigations of the quantum Hall effect in InxGa1–xAs-InP heterojunctions
,”
Phys. Rev. B
27
,
6549
(R) (
1983
).
31.
D. G.
Polyakov
and
B. I.
Shklovskii
, “
Conductivity peak broadening in the quantum Hall regime
,”
Phys. Rev. B
48
,
11167
(
1993
).
32.
H. P.
Wei
,
D. C.
Paalanen
, and
A. M. M.
Pruisken
,
Phys. Rev. Lett.
61
,
1294
(
1988
).
33.
H. P.
Wei
,
S. Y.
Lin
,
D. C.
Tsui
, and
A. M. M.
Pruisken
,
Phys. Rev. B
45
,
3926
(
1992
).
34.
S.
Koch
,
R. J.
Haug
,
K. v.
Klitzing
, and
K.
Ploog
,
Phys. Rev. B
46
,
1596
(
1992
).
35.
M.
Furlan
, “
Electronic transport and the localization length in the quantum Hall effect
,”
Phys. Rev. B
57
,
14818
(
1998
).
36.
F.
Hohls
,
U.
Zeitler
, and
R. J.
Haug
, “
Hopping conductivity in the quantum Hall effect: Revival of universal scaling
,”
Phys. Rev. Lett.
88
,
036802
(
2002
).
37.
K.
Bennaceur
,
P.
Jacques
,
F.
Portier
,
P.
Roche
, and
D. C.
Glattli
, “
Unveiling quantum Hall transport by Efros–Shklovskii to mott variable-range hopping transition in graphene
,”
Phys. Rev. B
86
,
085433
(
2012
).
38.
N. V.
Lien
and
R. L.
Rosenbaum
, “
General resistance crossover expressions for three-dimensional variable range hopping
,”
J. Phys.: Condens. Matter
10
,
6083
(
1998
).
39.
A.
Aharony
,
Y.
Zhang
, and
M. P.
Sarachik
,
Phys. Rev. Lett.
68
,
3900
(
1992
).
40.
G. S.
Boebinger
,
H. L.
Stormer
,
D. C.
Tsui
,
A. M.
Chang
,
J. C. M.
Hwang
,
A. Y.
Cho
,
C. W.
Tu
, and
G.
Weimann
, “
Activation energies and localization in the fractional quantum Hall effect
,”
Phys. Rev. B
36
,
7919
(
1987
).
41.
L.
Engel
,
H.
Wei
,
D.
Tsui
, and
M.
Shayegan
,
Surf. Sci.
229
,
13
(
1990
).
42.
T.
Machida
,
S.
Ishizuka
,
S.
Komiyama
,
K.
Muraki
, and
Y.
Hirayama
, “
International conference on high magnetic fields in semiconductors
,”
Physica B: Condens. Matter
298
,
182
(
2001
).
43.
P. T.
Madathil
,
K. A.
Villegas Rosales
,
C. T.
Tai
,
Y. J.
Chung
,
L. N.
Pfeiffer
,
K. W.
West
,
K. W.
Baldwin
, and
M.
Shayegan
,
Phys. Rev. Lett.
130
,
226503
(
2023
).
44.
Simrandeep
Kaur
,
Tanima
Chanda
,
Kazi Rafsanjani
Amin
,
Kenji
Watanabe
,
Takashi
Taniguchi
,
Unmesh
Ghorai
,
Yuval
Gefen
,
G. J.
Sreejith
, and
Aveek
Bid
, “
Universality of quantum phase transitions in the integer and fractional quantum hall regimes
,” arXiv:2312.06194 (2023).
45.
B. I.
Shklovskii
, “
Hopping conduction in semiconductors in a strong electric field
,”
Fiz. Tekh. Poluprov.
6
,
2335
(
1973
) [Sov. Phys.-Semicond. 6, 1964 (1973)].
46.
S.
Marianer
and
B. I.
Shklovskii
, “
Effective temperature of hopping electrons in a strong electric field
,”
Phys. Rev. B
46
,
13100
(
1992
).
47.
A. V.
Nenashev
,
J. O.
Oelerich
,
A. V.
Dvurechenskii
,
F.
Gebhard
, and
S. D.
Baranovskii
, “
Fundamental characteristic length scale for the field dependence of hopping charge transport in disordered organic semiconductors
,”
Phys. Rev. B
96
,
035204
(
2017
).
48.
M. M.
Fogler
and
R. S.
Kelley
, “
Non-ohmic variable-range hopping transport in one-dimensional conductors
,”
Phys. Rev. Lett.
95
,
166604
(
2005
).
49.
H. P.
Wei
,
L. W.
Engel
, and
D. C.
Tsui
, “
Current scaling in the integer quantum Hall effect
,”
Phys. Rev. B
50
,
14609
(
1994
).
50.
S. L.
Sondhi
,
S. M.
Girvin
,
J. P.
Carini
, and
D.
Shahar
,
Rev. Mod. Phys.
69
,
315
(
1997
).
51.
H.
Davies
,
P. A.
Lee
, and
T. M.
Rice
,
Phys. Rev. B
29
,
4260
(
1984
).
52.
A.
Mobius
,
M.
Richter
, and
B.
Drittler
,
Phys. Rev. B
45
,
11568
(
1992
).
53.
A.
Mobius
,
P.
Karmann
and
M.
Schreiber
,
J. Phys. Conf. Ser.
150
,
022057
(
2009
).
54.
A.
Mobius
and
M.
Richter
,
Phys. Rev. Lett.
105
,
039701
(
2010
).
55.
M.
Goethe
and
M.
Palassini
,
Phys. Rev. Lett.
103
,
045702
(
2009
).
56.
M.
Palassini
and
M.
Goethe
,
J. Phys.: Confer. Ser.
376
,
012009
(
2012
).
57.
A. L.
Efros
,
B.
Skinner
, and
B. I.
Shklovskii
, “
Coulomb gap in the one-particle density of states in three-dimensional systems with localized electrons
,”
Phys. Rev. B
84
,
064204
(
2011
).
58.
D. N.
Tsygankov
and
A. L.
Efros
, “
Variable range hopping in two-dimensional systems of interacting electrons
,”
Phys. Rev. Lett.
88
,
176602
(
2002
).
59.
E. I.
Levin
,
V. L.
Nguen
,
B. I.
Shklovskii
, and
A. L.
Efros
, “
Coulomb gap and hopping electric conduction. computer simulation
,”
Zh. Eksp. Teor. Fiz.
92
,
1499
(
1987
) [Sov. Phys.-JETP 65, 842 (1987)].
60.
M.
Sarvestani
,
M.
Schreiber
, and
T.
Vojta
, “
Coulomb gap at finite temperatures
,”
Phys. Rev. B
52
,
R3820
(
1995
).
61.
B. I.
Shklovskii
, “
Variable range hopping in thin film with large dielectric constant
,”
Fiz. Nizk. Temp.
43
,
902
(
2017
) [
Low Temp. Phys.
43, 699 (2017)].
62.
I. S.
Beloborodov
,
A. V.
Lopatin
,
V. M.
Vinokur
, and
K. B.
Efetov
, “
Granular electronic systems
,”
Rev. Mod. Phys.
79
,
469
(
2007
).
63.
J.
Zhang
and
B. I.
Shklovskii
, “
Density of states and conductivity of granular metal or array of quantum dots
,”
Phys. Rev. B
70
,
115317
(
2004
).
64.
T.
Chen
,
B.
Skinner
, and
B. I.
Shklovskii
, “
Coulomb gap triptych in a periodic array of metal nanocrystals
,”
Phys. Rev. Lett.
109
,
126805
(
2012
).
65.
Tianran
Chen
,
Brian
Skinner
, and
B. I.
Shklovskii
, “
Coulomb gap triptychs, square root of two effective charge, and hopping transport in periodic arrays of superconductor grains
,”
Phys. Rev. B
86
,
045135
(
2012
).
66.
J.
Mitchell
,
A.
Gangopadhyay
,
V.
Galitski
, and
M.
Müller
, “
Two-component Coulomb glass in insulators with a local attraction
,”
Phys. Rev. B
85
,
195141
(
2012
).
67.
T.
Hu
and
B. I.
Shklovskii
, “
Hopping conductivity of a suspension of nanowires in an insulator
,”
Phys. Rev. B
74
,
054205
(
2006
).
68.
T.
Hu
and
B. I.
Shklovskii
, “
Hopping conductivity of a suspension of flexible wires in an insulator
,”
Phys. Rev. B
74
,
174201
(
2006
).
69.
B.
Skinner
,
T.
Chen
, and
B. I.
Shklovskii
, “
Theory of hopping conduction in arrays of doped semiconductor nano-crystals
,”
Phys. Rev. B
85
,
205316
(
2012
).
70.
H.
Fu
,
K. V.
Reich
, and
B. I.
Shklovskii
, “
Hopping conductivity and insulator-metal transition in films of touching semiconductor nanocrystals
,”
Phys. Rev. B
93
,
125430
(
2016
).
71.
T.
Chen
,
K. V.
Reich
,
N. J.
Kramer
,
H.
Fu
,
U. R.
Kortshagen
, and
B. I.
Shklovskii
, “
Metal-insulator transition in doped semiconductor nanocrystal films
,”
Nature Mater.
15
,
299
(
2016
).
72.
M. M.
Fogler
,
S.
Teber
, and
B. I.
Shklovskii
, “
Variable-range hopping in quasi-one-dimensional electron crystals
,”
Phys. Rev. B
69
,
035413
(
2004
).
73.
M.
Bacani
,
M.
Novaka
,
I.
Kokanovic
, and
D.
Babic
, “
Hopping electron transport in doped polyaniline: An experimental verification of the Fogl–Teber–Shklovskii model
,”
Synthetic Metals
172
,
28
(
2013
).
74.
B. I.
Shklovskii
, “
New problem of percolation theory and conduction of granular metals
,”
Fiz. Tv. Tela
26
,
585
(
1984
).
75.
M. V.
Feigel’man
and
A. S.
Ioselevich
, “
Variable-range cotunneling and conductivity of a granular metal
,”
JETP. Lett.
81
,
277
(
2005
).
76.
I. M.
Lifshitz
and
V. Ya.
Kirpichenkov
, “
Tunnel transparency of disordered systems
,”
Zh. Eksp. Theor. Fiz.
77
,
989
(
1979
) [Sov. Phys.-JETP 50, 499 (1979)].
77.
B. I.
Shklovskii
and
A. L.
Efros
, “
Tunnel transparency of disordered systems in a magnetic field
,”
Zh. Eksp. Theor. Fiz.
84
,
81
(
1983
) [Sov. Phys.-JETP 57, 470 (1983)].
78.
A. B.
Khaetskii
and
B. I.
Shklovskii
, “
Anomalous tunneling across magnetic field near semiconductor surface
,”
Zh. Eksp. Theor. Fiz.
85
,
721
(
1983
) [Sov. Phys.-JETP 58, 421 (1983)].
79.
B. Z.
Spivak
,
V. A.
Kharchenko
, and
B. I.
Shklovskii
, “
Scattering effect in VRH conduction
,”
Fiz. Tekh. Poluprov.
19
,
493
(
1985
).
80.
V. L.
Nguyen
,
B. Z.
Spivak
, and
B. I.
Shklovskii
, “
Tunnel hops in disordered systems
,”
Zh. Eksp.Theor. Fiz.
89
,
1770
(
1985
) [Sov. Phys.-JETP 62, 1021 (1985)].
81.
V. L.
Nguyen
,
B. I.
Shklovskii
, and
B. Z.
Spivak
, “
Aharonov–bohm oscillations with normal and superconductive flux-quanta in hopping conduction
,”
Pis’ma Zh. Eksp. Teor. Fiz.
41
,
35
(
1985
) [Sov. Phys. JETP Lett. 41, 42 (1985)].
82.
B. I.
Shklovskii
and
B. Z.
Spivak
, “
Interference phenomena in variable range hopping conductivity
,” in
Hopping and Related Phenomena
, edited by
H.
Fritzsche
and
M.
Pollak
(
World Scientific
,
1990
).
83.
B. I.
Shklovskii
and
B. Z.
Spivak
, “
Scattering and interference phenomena in variable range hopping conduction
,” in
Hopping Conduction in Semiconductors
, edited by
M.
Pollak
and
B. I.
Shklovskii
(
North-Holland
,
Amsterdam
,
1990
).
84.
M. M.
Fogler
,
A. Yu.
Dobin
, and
B. I.
Shklovskii
, “
Localization length at the resistivity minima of the quantum Hall effect
,”
Phys. Rev. B
57
,
4614
(
1998
).
85.
I. P.
Zvyagin
,
Phys. Status Solidi B
58
,
443
(
1973
).
86.
M. J.
Burns
and
P. M.
Chaikin
, “
Interaction effects and thermoelectric power in low-temperature hopping
,”
J. Phys. C: Solid State Phys.
18
,
L743
(
1985
).
87.
T.
Yamamoto
,
M.
Ogata
, and
H.
Fukuyama
, “
Thermoelectric effect in mott variable-range hopping
,”
J. Phys. Soc. Jpn.
91
,
044704
(
2022
).
88.
X.
Liu
,
A.
Sidorenko
,
S.
Wagner
,
P.
Ziegler
, and
H. v.
Löhneysen
, “
Electronic transport processes in heavily doped uncompensated and compensated silicon as probed by the thermoelectric power
,”
Phys. Rev. Lett.
77
,
3395
(
1996
).
89.
M.
Park
,
S. J.
Hong
,
K. H.
Kim
,
H.
Kang
,
M.
Lee
,
D. H.
Jeong
, and
Y. W.
Park
, and
B. H.
Kim
, “
Electrical and thermoelectric transport by variable range hopping in reduced graphene oxide
,”
Appl. Phys. Lett.
111
,
173103
(
2017
).
90.
K.
Kanahashi
,
Y.-Y.
Noh
,
W.-T.
Park
, and
H.
Yang
,
Hiromichi Ohta Hisaaki
Tanaka
, and
Taishi
Takenobu
, “
Charge and thermoelectric transport mechanism in donor-acceptor copolymer films
,”
Phys. Rev. Res.
2
,
043330
(
2020
).
91.
Mark
Lee
,
J. G.
Massey
,
V. L.
Nguyen
, and
B. I.
Shklovskii
, “
Coulomb gap in a doped semiconductor near the metal-insulator transition: Tunneling experiment and scaling ansatz
,”
Phys. Rev. B
60
,
1582
(
1999
) (misprints are corrected in: arXiv:cond-mat/9902169).
92.
W. L.
McMillan
, “
Scaling theory of the metal-insulator transition in amorphous materials
,”
Phys. Rev. B
24
,
2739
(
1981
).
93.
M.
Amini
,
V. E.
Kravtsov
, and
M.
Mueller
, “
Multifractality and quantum to classical crossover in the Coulomb anomaly at the Mott–Anderson metal-insulator transition
,”
New J. Phys.
16
,
015022
(
2014
).
94.
I. S.
Burmistrov
,
I. V.
Gornyi
, and
A. D.
Mirlin
, “
Tunneling into the localized phase near anderson transitions with Coulomb interaction
,”
Phys. Rev. B
89
,
035430
(
2014
).
95.
B. L.
Al’tshuler
and
A. G.
Aronov
, “
Contribution to the theory of disordered metals in heavily doped semiconductors
,”
Zh. Eksp. Theor. Fiz.
77
,
2028
(
1979
) [Sov. Phys.-JETP 50, 968 (1979)].
96.
B. L.
Al’tshuler
and
A. G.
Aronov
, “
Electron-electron interaction in disordered conductors
,” in
Electron-Electron Interaction in Disordered Systems
, edited by
A. L.
Efros
and
M.
Pollak
(
North-Holland
,
Amsterdam
,
1985
).
97.
W. L.
McMillan
and
J.
Mochel
,
Phys. Rev. Lett.
40
,
556
(
1981
).
98.
S. D.
Baranovskii
,
B. I.
Shklovskii
, and
A. L.
Efros
, “
Screening in localized electron system
,”
Zh. Eksp. Teor. Fiz.
87
,
1793
(
1984
) [Sov. Phys. JETP 60, 1031 (1984)].
99.
S.-R.
Eric Yang
,
Ziqiang
Wang
, and
A. H.
MacDonald
, “
Thermodynamic and tunneling density of states of the integer quantum Hall critical state
,”
Phys. Rev. B
65
,
041302
(R) (
2001
).
100.
V. Yu.
Butko
,
J. F.
DiTusa
, and
P. W.
Adams
, “
Coulomb gap: How a metal film becomes an insulator
,”
Phys. Rev. Lett.
84
,
1543
(
2000
).
101.
Y. H.
Song
,
Z. Y.
Jia
,
D.
Zhang
,
X. Y.
Zhu
,
Z. Q.
Shi
,
H.
Wang
,
L.
Zhu
,
Q. Q.
Yuan
,
H.
Zhang
,
D. Y.
Xing
, and
S. C.
Li
,
Nat. Commun.
9
,
4071
(
2018
).
102.
B. H.
Moon
,
J. J.
Bae
,
M.-K.
Joo
,
H.
Choi
,
G. H.
Han
,
H.
Lim
, and
Y. H.
Lee
,
Nat. Commun.
9
,
2052
(
2018
).
103.
N. F.
Mott
,
Philos. Mag.
22
,
7
(
1970
);
N. F.
Mott
and
E. A.
Davis
,
Electronic Processes in Non-Crystalline Materials
(Clarendon Press,
Oxford
,
1971
).
104.
V. L.
Berezinskii
,
Zh. Eksp. Teor. Fiz.
65
,
125
(
1973
) [Sov. Phys. JETP 38, 620 (1974)].
105.
B. I.
Shklovskii
and
A. L.
Efros
, “
Phononless hopping conduction in disordered systems
,”
Zh. Eksp. Theor. Fiz.
81
,
406
(
1981
) [Sov. Phys.-JETP 54, 218 (1981)].
106.
R. M.
Lewis
and
J. P.
Carini
, “
Frequency scaling of microwave conductivity in the integer quantum Hall effect minima
,”
Phys. Rev. B
64
,
073310
(
2000
).
107.
F.
Hohls
,
U.
Zeitler
, and
R. J.
Haug
, “
High frequency conductivity in the quantum Hall regime
,”
Phys. Rev. Lett.
86
,
5124
(
2001
).
108.
M.
Lee
and
M. L.
Stutzmann
, “
Microwave ac conductivity spectrum of a Coulomb glass
,”
Phys. Rev. Lett.
87
,
056402
(
2001
).
109.
E.
Helgren
,
N. P.
Armitage
, and
G.
Gruner
, “
Electrodynamics of a Coulomb glass in n-type silicon
,”
Phys. Rev. Lett.
89
,
246601
(
2002
).
110.
E.
Helgren
,
N. P.
Armitage
, and
G.
Gruner
, “
Frequency-dependent conductivity of electron glasses
,”
Phys. Rev. B
69
,
014201
(
2004
).
111.
M.
Hering
,
M.
Scheffler
,
M.
Dressel
, and
H. v.
Löhneysen
,
Phys. Rev. B
75
,
205203
(
2007
).
112.
M.
Pollak
and
T. H.
Geballe
,
Phys. Rev.
122
,
1742
(
1961
).
113.
I. G.
Austin
and
N. F.
Mott
,
Adv. Phys.
18
,
41
(
1969
).
115.
A. L.
Efros
and
B. I.
Shklovskii
, “
Electron-electron interaction in disordered systems with localized states
,” in
Electron-Electron Interaction in Disordered Systems
, edited by
A. L.
Efros
and
M.
Pollak
(
North-Holland
,
Amsterdam
,
1985
).
116.
S. D.
Baranovskii
,
A. A.
Uzakov
, and
A. L.
Efros
, “
Thermodynamic properties of impurity band electrons
,”
Zh. Eksp. Theor. Fiz.
83
,
756
(
1982
) [Sov. Phys.- JETP 56, 422 (1982)].
117.
I. P.
Zvyagin
,
Phys. Status Solidi B
97
,
143
(
1980
).
118.
H.
Bottger
,
V.
Bryksin
, and
G. Yu.
Yashin
,
J. Phys. C
12
,
3951
(
1979
).
119.
B. I.
Shklovskii
, “
Theory of 1/f noise in hopping conduction
,”
Solid State Commun.
33
,
273
(
1980
).
120.
B. I.
Shklovskii
, “
1/f noise in variable range hopping conduction
,”
Phys. Rev. B
67
,
045201
(
2003
).
121.
A. L.
Burin
,
Yu. M.
Galperin
,
V.
Kozub
,
V.
Vinokur
, and
B. I.
Shklovskii
, “
Many-electron theory of 1/f-noise in hopping conductivity
,”
Phys. Rev. B
74
,
075205
(
2006
).
122.
B. I.
Shklovskii
, “
Non-ohmic hopping conduction
,”
Fiz. Tekh. Poluprov.
10
,
1440
(
1976
) [Sov. Phys.-Semicond. 10, 855 (1976)].
123.
B. I.
Shklovskii
, “
Percolation conductivity in strong electric fields
,”
Fiz. Tekh. Poluprov.
13
,
93
(
1979
) [Sov. Phys.-Semicond. 13, 53 (1979)].
124.
E. I.
Levin
, “
Percolative nonohmic conduction in polycrystalline semiconductors
,”
Fiz. Tekh. Poluprov.
17
,
255
(
1983
) [Sov. Phys.-Semicond. 17, 158 (1983)].
125.
E. I.
Levin
,
N.
Van Lien
, and
B. I.
Shklovskii
, “
Hopping electrical conduction in strong electric fields: A numerical computer experiment
,”
Fiz. Tekh. Poluprov.
16
,
815
(
1982
) [Sov. Phys. Semicond. 16, 523 (1982)].
126.
E. I.
Levin
and
B. I.
Shklovskii
, “
Low-temperature hopping conduction in strong electric field, computer experiment
,”
Fiz. Tekh. Poluprov.
18
,
856
(
1984
) [Sov. Phys. Semicond. 11, 534 (1984)].
127.
E. I.
Levin
,
I. M.
Ruzin
, and
B. I.
Shklovskii
, “
Transverse hopping conductivity of amorphous films in strong electric fields
,”
Fiz. Tekh. Poluprov.
22
,
624
(
1988
) [Sov. Phys.-Semicond. 22, 401 (1988)].
128.
N.
Van Lien
and
B. I.
Shklovskii
, “
Hopping conduction in strong electric fields and directed percolation
,”
Solid State Commun.
38
,
99
(
1981
).
129.
E. I.
Levin
and
B. I.
Shklovskii
, “
Negative differential conductivity of low density electron gas in random potential
,”
Solid State Commun.
67
,
233
(
1988
).
130.
D. I.
Aladashvili
,
Z.
Adamia
,
K.
Lavdovskii
,
E. I.
Levin
, and
B. I.
Shklovskii
, “
Negative differential resistance in hopping region in silicon
,”
Pis’ma Zh. Eksp. Teor. Fiz.
47
,
390
(
1988
) [Sov. Phys.-JETP Lett. 47, 466 (1988)].
131.
A. V.
Nenashev
,
F.
Jansson
,
S. D.
Baranovskii
,
R.
Osterbacka
,
A. V.
Dvurechenskii
, and
F.
Gebhard
, “
Hopping conduction in strong electric fields: Negative differential conductivity
,”
Phys. Rev. B
78
,
165207
(
2008
).
132.
B. I.
Shklovskii
, “
Model of surface recombination
,”
Pis’ma Zh. Eksp. Theor. Fiz.
44
,
95
(
1986
) [Sov. Phys.- JETP Letters 44, 123 (1986)].
133.
S. D.
Baranovskii
,
V. G.
Karpov
, and
B. I.
Shklovskii
, “
Theory of non-radiative recombination in amorphous semiconductors
,”
J. Non-Cryst. Solids
97
,
487
(
1987
).
134.
B. I.
Shklovskii
,
H.
Fritzsche
, and
S. D.
Baranovskii
, “
Electronic transport and recombination in amorphous semi-conductors at low temperatures
,”
Phys. Rev. Lett.
62
,
2989
(
1989
).
135.
S. D.
Baranovskii
,
H.
Fritzsche
,
E. I.
Levin
,
I. M.
Ruzin
, and
B. I.
Shklovskii
, “
Low temperature electronic transport and recombination in amorphous semiconductors
,”
Zh. Eksp. Teor. Fiz.
96
,
1362
(
1989
) [Sov. Phys. JETP 69, 773 (1990)].
136.
S. D.
Baranovskii
,
E. I.
Ivchenko
, and
B. I.
Shklovskii
, “
New regime of recombination of photoinduced carriers in amorphous semiconductors
,”
Zh. Eksp. Theor. Fiz.
22
,
2234
(
1987
) [Sov. Phys.-JETP 65, 1260 (1987)].
137.
I. M.
Ruzin
and
B. I.
Shklovskii
, “
Theory of hopping photoconductivity at low-frequency excitation
,”
Fiz. Tekh. Poluprov.
23
,
1881
(
1989
) [Sov. Phys.- Semicond. 23, 1164 (1989)].
138.
B. I.
Shklovskii
, “
Dyakonov–Perel spin relaxation near metal-insulator transition and in hopping transport
,”
Phys. Rev. B
73
,
193201
(
2006
).
139.
L. I.
Glazman
,
I. M.
Ruzin
, and
B. I.
Shklovskii
, “
Quantum transport and pinning of one-dimensional Wigner crystal
,”
Phys. Rev. B
45
,
8454
(
1992
).
140.
I. M.
Ruzin
,
S.
Marianer
, and
B. I.
Shklovskii
, “
Pinning of two-dimensional wigner crystal by charged impurities
,”
Phys. Rev. B
46
,
3999
(
1992
).
141.
B. I.
Shklovskii
, “
Coulomb gap and variable range hopping in a pinned Wigner crystal
,”
Phys. Status Solidi C
1
,
46
(
2004
).
142.
A. A.
Koulakov
and
B. I.
Shklovskii
, “
Charging spectrum and configurations of a Wigner crystal island
,”
Phys. Rev. B
57
,
2352
(
1998
).
143.
H. L.
Stormer
,
J. C.
Weiner
,
L. N.
Pfeiffer
,
K. W.
Baldwin
,
S. J.
Pearton
, and
K. W.
West
,
Phys. Rev. Lett.
68
,
3088
(
1992
);
R. C.
Ashoori
,
H. L.
Stormer
,
J. C.
Weiner
,
L. N.
Pfeiffer
,
K. W.
Baldwin
, and
K. W.
West
,
Phys. Rev. Lett.
71
,
613
(
1993
).
144.
M. E.
Raikh
,
L. I.
Glazman
, and
L. E.
Zhukov
,
Phys. Rev. Lett.
77
,
1354
(
1996
).
145.
U.
Sivan
,
R.
Berkovits
,
Y.
Aloni
,
O.
Prus
,
A.
Auerbach
, and
G.
Ben-Yoseph
,
Phys. Rev. Lett.
77
,
1123
(
1996
).
146.
A. A.
Koulakov
,
F. G.
Pikus
, and
B. I.
Shklovskii
, “
Statistics of the charging spectrum of a two-dimensional Coulomb-glass island
,”
Phys. Rev. B
55
,
9223
(
1997
).
147.
Brian
Skinner
,
Tianran
Chen
,
M. S.
Loth
, and
B. I.
Shklovskii
, “
Theory of volumetric capacitance of electric double layer capacitor
,”
Phys. Rev. E
83
,
056102
(
2011
).
148.
B.
Skinner
,
T.
Chen
, and
B. I.
Shklovskii
, “
Cooperative charging in a nanocrystal assembly gated by ionic liquid
,”
Phys. Rev. B
84
,
245304
(
2011
).
149.
M. M.
Fogler
and
B. I.
Shklovskii
, “
Collapse of spin-splitting in the quantum Hall effect
,”
Phys. Rev. B
52
,
17366
(
1995
).
150.
I. L.
Aleiner
and
L. I.
Glazman
,
Phys. Rev. B
52
,
296
(
1995
).
151.
A. A.
Koulakov
,
M. M.
Fogler
, and
B. I.
Shklovskii
, “
Charge density wave in two-dimensional electron liquid in a weak magnetic field
,”
Phys. Rev. Lett.
76
,
499
(
1996
).
152.
A. A.
Koulakov
,
M. M.
Fogler
, and
B. I.
Shklovskii
, “
The ground state of a two-dimensional electron liquid in a weak magnetic field
,”
Phys. Rev. B
54
,
1853
(
1996
).
153.
M. P.
Lilly
,
K. B.
Cooper
,
J. P.
Eisenstein
,
L. N.
Pfeiffer
, and
K. W.
West
, “
Evidence for an anisotropic state of two-dimensional electrons in high Landau levels
,”
Phys. Rev. Lett.
82
,
394
(
1999
).
154.
R. R.
Du
,
D. C.
Tsui
,
H. L.
Stormer
,
L. N.
Pfeiffer
,
K. W.
Baldwin
, and
K. W.
West
, “
Strongly anisotropic transport in higher two-dimensional Landau levels
,”
Solid State Commun.
109
,
389
(
1999
).
155.
M. M.
Fogler
,
Stripe and bubble phases in quantum Hall systems, high magnetic fields
, in:
Applications in Condensed Matter Physics and Spectroscopy
(
Springer-Verlag
,
Berlin
,
2002
); arXiv:cond-mat/0111001.
156.
A. H.
MacDonald
and
M. P. A.
Fisher
,
Phys. Rev. B
61
,
5724
(
2000
).
157.
M.
Sammon
,
X.
Fu
,
Yi
Huang
,
M. A.
Zudov
,
B. I.
Shklovskii
,
G. C.
Gardner
,
J. D.
Watson
,
M. J.
Manfra
,
K. W.
Baldwin
,
L. N.
Pfeiffer
, and
K. W.
West
, “
Resistivity anisotropy of quantum Hall stripe phases
,”
Phys. Rev. B
100
,
241303
(R) (
2019
).
158.
Y.
Huang
,
M.
Sammon
,
M. A.
Zudov
, and
B. I.
Shklovskii
, “
Isotropically conducting (hidden) quantum Hall stripe phases in a two-dimensional electron gas
,”
Phys. Rev. B
101
,
161302
(
2020
).
159.
X.
Fu
,
Yi
Huang
,
M. A.
Zudov
,
B. I.
Shklovskii
,
G. C.
Gardner
, and
M. J.
Manfra
, “
Hidden quantum Hall stripes in AlxGa1–xAs/Al0.24 Ga0.76As quantum wells
,”
Phys. Rev. Lett.
125
,
236803
(
2020
).
160.
X.
Fu
,
Q.
Shi
,
M. A.
Zudov
,
G. C.
Gardner
,
J. D.
Watson
, and
M. J.
Manfra
, “
Two- and three-electron bubbles in AlxGa1–xAs/Al0.24 Ga0.76As quantum wells
,”
Phys. Rev. B
99
,
161402
(
2019
).
161.
D.
Ro
,
N.
Deng
,
J. D.
Watson
,
M. J.
Manfra
,
L. N.
Pfeiffer
,
K. W.
West
, and
G. A.
Csáthy
, “
Electron bubbles and the structure of the orbital wave function
,”
Phys. Rev. B
99
,
201111
(
2019
).
162.
S.
Chen
,
R.
Ribeiro-Palau
,
K.
Yang
,
K.
Watanabe
,
T.
Taniguchi
,
J.
Hone
,
M. O.
Goerbig
, and
C. R.
Dean
, “
Competing fractional quantum Hall and electron solid phases in graphene
,”
Phys. Rev. Lett.
122
,
026802
(
2019
).
163.
Fangyuan
Yang
,
Ruiheng
Bai
,
Alexander A.
Zibrov
,
Sandeep
Joy
,
Takashi
Taniguchi
,
Kenji
Watanabe
,
Brian
Skinner
,
Mark O.
Goerbig
, and
Andrea F.
Young
, “
Cascade of multi-electron bubble phases in monolayer graphene at high Landau level filling
,” arXiv:2309.04319 (2023).
164.
M. S.
Bello
,
E. I.
Levin
,
B. I.
Shklovskii
, and
A. L.
Efros
, “
Density of localized states in the surface impurity band of a metal-insulator-semiconductor structure
,”
Zh. Eksp. Theor. Fiz.
83
,
1596
(
1981
) [Sov. Phys.-JETP 53, 822 (l981)].
165.
E. V.
Chenskii
and
Yu. Ya.
Tkach
,
Zh. Eksp. Fiz.
79
,
1809
(
1980
) [Sov. Phys. JETP 52, 915 (1980)].
166.
S. V.
Kravchenko
,
D. A.
Rinberg
,
S. G.
Semenchinsky
, and
V. M.
Pudalov
,
Phys. Rev. B
42
,
3741
(
1990
).
167.
J. P.
Eisenstein
,
L. N.
Pfeiffer
, and
K. W.
West
,
Phys. Rev. Lett.
68
,
674
(
1992
).
168.
S.
Shapira
,
U.
Sivan
,
P. M.
Solomon
,
E.
Buchstab
,
M.
Tischler
, and
G.
Ben Yoseph
,
Phys. Rev. Lett.
77
,
3181
(
1996
).
169.
S. C.
Dultz
and
H. W.
Jiang
,
Phys. Rev. Lett.
84
,
4689
(
2000
).
170.
S.
Ilani
,
A.
Yacoby
,
D.
Mahalu
, and
H.
Shtrikman
,
Phys. Rev. Lett.
84
,
3133
(
2000
).
171.
G.
Allison
,
E. A.
Galaktionov
,
A. K.
Savchenko
,
S. S.
Safonov
,
M. M.
Fogler
,
M. Y.
Simmons
, and
D. A.
Ritchie
,
Phys. Rev. Lett.
96
,
216407
(
2006
).
172.
B.
Tanatar
and
D. M.
Ceperley
, “
Ground state of the two-dimensional electron gas
,”
Phys. Rev. B
39
,
5005
(
1989
).
173.
B.
Skinner
and
B. I.
Shklovskii
, “
Anomalously large capacitance of a plane capacitor with a two-dimensional electron gas
,”
Phys. Rev. B
82
,
155111
(
2010
).
174.
L.
Li
,
C.
Richter
,
S.
Paetel
,
T.
Kopp
,
J.
Mannhart
, and
R. C.
Ashoori
,
Science
332
,
825
(
2011
).
175.
T.
Li
,
J.
Zhu
,
Y.
Tang
,
K.
Watanabe
,
T.
Taniguchi
,
V.
Elser
, and
J.
Shan
, and
Kin Fai
Mak
, “
Charge-order-enhanced capacitance in semiconductor moiré superlattices
,”
Nature Nanotechn.
16
,
1068
(
2021
).
176.
A. L.
Efros
, in
Proceedings of 20th International Conference on the Physics of Semiconductors, Thessaloniki, 1990
, edited by
E. N.
Anastassakis
and
J. D.
Joannopoulos
, (
World Scientific
,
Singapore
,
1990
), Vol. 1, pp.
59
66
.
177.
B.
Skinner
and
B. I.
Shklovskii
,
Phys. Rev. B
87
,
035409
(
2013
).
178.
B.
Skinner
,
G. L.
Yu
,
A. V.
Kretinin
,
A. K.
Geim
,
K. S.
Novoselov
, and
B. I.
Shklovskii
, “
Effect of dielectric response on the quantum capacitance of graphene in a strong magnetic field
,”
Phys. Rev. B
88
,
155417
(
2013
).
179.
B. I.
Shklovskii
, “
Screening of a macroion by multivalent ions: Correlation-induced inversion of charge
,”
Phys. Rev. E
60
,
5802
(
1999
).
180.
T. T.
Nguyen
,
A. Yu.
Grosberg
, and
B. I.
Shklovskii
, “
Colloquium: Physics of charge inversion in chemical and biological systems
,”
Rev. Mod. Phys.
74
,
329
(
2002
).
181.
M. S.
Loth
and
B. I.
Shklovskii
, “
Non-mean-field screening by multivalent counterions
,”
J. Phys.: Condens. Matter
21
,
424104
(
2009
).
182.
T. T.
Nguyen
and
B. I.
Shklovskii
, “
Inversion of DNA charge by a positive polymer via fractionalization of the polymer charge
,”
Physica A
310
,
197
(
2002
).
184.
A. V.
Kabanov
and
V. A.
Kabanov
,
Adv. Drug Delivery Rev.
30
,
49
(
1998
).
185.
X.
Hou
,
T.
Zaks
,
R.
Langer
, and
Y.
Dong
, “
Lipid nanoparticles for mRNA delivery
,”
Nature Rev. Mater.
6
,
1078
(
2021
).
186.
Tao
Hu
and
B. I.
Shklovskii
, “
Kinetics of viral self-assembly: Role of the single-stranded RNA antenna
,”
Phys. Rev. E
75
,
051901
(
2007
).
187.
A.
Kamenev
,
J.
Zhang
,
B. I.
Shklovskii
, and
A. I.
Larkin
, “
Transport in one dimensional Coulomb gas: From ion channels to nanopores
,”
Physica A
359
,
129
(
2005
).
188.
A.
Kamenev
,
J.
Zhang
, and
B. I.
Shklovskii
, “
Conductance of ion channels and nanopores with charged walls: A toy model
,”
Phys. Rev. Lett.
95
,
148101
(
2005
).
189.
L.
Susskind
, “
Lattice models of quark confinement at high temperature
,”
Phys. Rev. D
20
,
2610
(
1979
).
190.
E. V.
Gorbar
,
V. P.
Gusynin
,
D. O.
Oriekhov
, and
B. I.
Shklovskii
, “
Electron binding energy of donors in bilayer graphene with a gate-tunable gap
,”
Phys. Rev. B
109
,
165145
(
2024
).
191.
B. I.
Shklovskii
,
B.
Shapiro
,
B. R.
Sears
,
P.
Lambrianidis
, and
H. B.
Shore
, “
Statistics of spectra of disordered systems near the metal-insulator transition
,”
Phys. Rev. B
47
,
11487
(
1993
).
192.
I. Kh.
Zharekeshev
and
B.
Kramer
, “
Asymptotics of universal probability of neighboring level spacings at the Anderson transition
,”
Phys. Rev. Lett.
79
,
717
(
1997
).
193.
V.
Oganesyan
and
D. A.
Huse
, “
Localization of interacting fermions at high temperature
,”
Phys. Rev. B
75
,
155111
(
2007
).
194.
F. J.
Dyson
,
J. Math. Phys.
3
,
140
(
1962
).
F. J.
Dyson
and
M. L.
Mehta
,
J. Math. Phys.
4
,
701
(
1963
).
195.
B. L.
Al’tshuler
and
B. I.
Shklovskii
, “
Repulsion of energy levels and conductivity of small metal samples
,”
Zh. Eksp. Teor. Fiz.
91
,
220
(
1986
) [Sov. Phys. JETP 64, 127 (1986)].
196.
M. M.
Fogler
and
B. I.
Shklovskii
, “
Probability of an eigenvalue number fluctuation in an interval of a random matrix spectrum
,”
Phys. Rev. Lett.
74
,
3312
(
1995
).
197.
B. L.
Altshuler
,
I.
Zharekeshev
,
S. A.
Kotochigova
, and
B. I.
Shklovskii
, “
Repulsion between energy levels and the metal-insulator transition
,”
Zh. Eksp. Teor. Fiz.
94
,
343
(
1988
) [Sov. Phys.-JETP 67, 625 (1988)].
198.
A. F.
Tzortzakakis
,
K. G.
Makris
, and
E. N.
Economou
,
Phys. Rev. B
101
,
014202
(
2020
).
199.
L.
,
P.
Ribeiro
, and
T.
Prosen
, “
Complex spacing ratios: A signature of dissipative quantum chaos
,”
Phys. Rev. X
10
,
021019
(
2020
).
200.
Y.
Huang
and
B. I.
Shklovskii
, “
Anderson transition in three-dimensional non-Hermitian disorder
,”
Phys. Rev. B
101
,
014204
(
2020
).
201.
Y.
Huang
and
B. I.
Shklovskii
, “
Spectral rigidity of non-Hermitian symmetric random matrices
,”
Phys. Rev. B
102
,
064212
(
2020
).
202.
M. M.
Fogler
,
A. Yu.
Dobin
,
V. I.
Perel
, and
B. I.
Shklovskii
,
Phys. Rev. B
56
,
6823
(
1997
).
203.
A. D.
Mirlin
,
J.
Wilke
,
F.
Evers
,
D. G.
Polyakov
, and
P.
Wolfle
, “
Strong magnetoresistance induced by long-range disorder
,”
Phys. Rev. Lett.
83
,
2801
(
1999
).
204.
Yu. Ya.
Tkach
and
E. V.
Chenskii
,
Zh. Eksp. Fiz.
102
,
1683
(
1992
) [Sov. Phys. JETP 75, 910 (1992)].
205.
J. J.
Nelson
,
K. V.
Reich
,
M.
Sammon
,
B. I.
Shklovskii
, and
A. M.
Goldman
, “
Hopping conduction via ionic liquid induced silicon surface states
,”
Phys. Rev. B
92
,
085524
(
2015
).
206.
Yu. M.
Gal’perin
,
E. P.
German
, and
V. G.
Karpov
, “
Hall effect under hopping conduction conditions
,”
Zh. Eksp. Teor. Fiz.
99
,
343
(
1991
) [Sov. Phys. JETP 72, 194 (1991)].
207.
D.
Stauffer
and
A.
Aharony
,
Introduction to Percolation Theory
(
Taylor and Francis
,
1994
).
208.
P. P.
Orth
,
R. M.
Fernandes
,
J.
Walter
,
C.
Leighton
, and
B. I.
Shklovskii
, “
Percolation via combined electrostatic and chemical doping in complex oxide films
,”
Phys. Rev. Lett.
118
,
106801
(
2017
).
209.
Jeff
Walter
,
T.
Charlton
,
H.
Ambaye
,
M. R.
Fitzsimmons
,
Peter P.
Orth
,
R. M.
Fernandes
, and
Chris
Leighton
, “
Giant electrostatic modification of magnetism via electrolyte-gate-induced cluster percolation in La1–xSrxCoO3–δ
,”
Phys. Rev. Mat.
2
,
111406
(
2018
).
210.
L. D.
Landau
and
E. M.
Lifshitz
,
Electrodynamics of Continuous Media
, Chapter II (
Pergamon Press
,
1960
).
211.
D. N.
Arnold
,
G.
David
,
D.
Jerison
,
S.
Mayboroda
, and
M.
Filoche
, “
Effective confining potential of quantum states in disordered media
,”
Phys. Rev. Lett.
116
,
056602
(
2016
).
212.
F.
Gebhard
,
A. V.
Nenashev
,
K.
Meyerholz
, and
S. D.
Baranovskii
, “
Quantum states in disordered media. I. Low-pass filter approach
,”
Phys. Rev. B
107
,
064206
(
2023
).
213.
B. I.
Shklovskii
and
A. L.
Efros
, “
Density of state oscillations of 2d electron gas in magnetic field
,”
Pis’ma Zh. Eksp. Theor. Fiz.
44
,
95
(
1986
) [Sov. Phys.-JETP Lett. 44, 669 (1986)].
214.
A. L.
Efros
,
Solid State Commun.
67
,
1019
(
1988
).
215.
A. L.
Efros
,
Solid State Commun.
65
,
1281
(
1988
).
216.
217.
D. B.
Chklovskii
,
B. I.
Shklovskii
and
L. I.
Glazman
, “
Electrostatics of edge states
,”
Phys. Rev. B
46
,
4026
(
1992
).
218.
D. B.
Chklovskii
,
K. A.
Matveev
, and
B. I.
Shklovskii
, “
Ballistic conductance of interacting electrons
,”
Phys. Rev. B
47
,
12605
(
1993
).
219.
M. M.
Fogler
and
B. I.
Shklovskii
, “
Resistance of a long wire in the quantum Hall regime
,”
Phys. Rev. B
50
,
1656
(
1994
).
220.
M.
Sammon
,
M. A.
Zudov
, and
B. I.
Shklovskii
, “
Mobility and quantum mobility of modern GaAs/AlGaAs heterostructures
,”
Phys. Rev. M
2
,
064604
(
2018
).
221.
M.
Sammon
,
T.
Chen
, and
B. I.
Shklovskii
, “
Excess electron screening of remote donors and mobility in modern GaAs/AlGaAs heterostructures
,”
Phys. Rev. Mater.
2
,
104001
(
2018
).
222.
B.
Skinner
,
T.
Chen
, and
B. I.
Shklovskii
, “
Why is the bulk resistivity of topological insulators so small?
,”
Phys. Rev. Lett.
109
,
176801
(
2012
).
223.
T.
Knispel
,
W.
Jolie
,
N.
Borgwardt
,
J.
Lux
,
Zhiwei
Wang
,
Yoichi
Ando
,
A.
Rosch
,
T.
Michely
, and
M.
Grüninger
, “
Charge puddles in the bulk and on the surface of the topological insulator BiSbTeSe2 studied by scanning tunneling microscopy and optical spectroscopy
,”
Phys. Rev. B
96
,
195135
(
2017
).
224.
B.
Skinner
and
B. I.
Shklovskii
, “
Theory of the random potential and conductivity at the surface of a topological insulator
,”
Phys. Rev. B
87
,
0755454
(
2013
).
225.
Y.
Huang
and
B. I.
Shklovskii
, “
Disorder effects in topological insulator thin films
,”
Phys. Rev. B
103
,
165409
(
2021
).
226.
Y.
Huang
and
B. I.
Shklovskii
, “
Disorder effects in topological insulator nanowires
,”
Phys. Rev. B
104
,
054205
(
2021
).
227.
S.
Adam
,
E. H.
Hwang
,
V. M.
Galitski
, and
S.
Das Sarma
, “
A self-consistent theory for graphene transport
,”
PNAS
104
,
18392
(
2007
).
228.
Y.
Huang
,
Y.
He
,
B.
Skinner
, and
B. I.
Shklovskii
, “
Conductivity of two-dimensional narrow gap semiconductors subjected to strong Coulomb disorder
,”
Phys. Rev. B
105
,
054206
(
2022
).
229.
Y.
Huang
,
B.
Skinner
, and
B. I.
Shklovskii
, “
Conductivity of two-dimensional small gap semiconductors and topological insulators in strong Coulomb disorder
,”
JETP
135
,
409
(
2022
).
230.
V. F.
Mitin
, “
Heavily doped and fully compensated Ge single crystalline films on GaAs
,”
Appl. Phys. Lett.
92
,
202111
(
2008
).
231.
D.
Rao
,
D. P.
Panda
,
A.
Indiradevi
,
K.
Pillai
,
A.
Tayal
,
M.
Garbrecht
, and
B.
Saha
, “
Quasiclassical anderson transition and thermally activated percolative charge transport in single-crystalline ScN
,”
Phys. Rev. B
109
,
155307
(
2024
).
232.
Yi
Huang
,
Y.
Ayino
, and
B. I.
Shklovskii
, “
Metal-insulator transition in n-type bulk crystals and films of strongly compensated SrTiO3
,”
Phys. Rev. Mater.
5
,
044606
(
2021
).
233.
M.
Mueller
and
B. I.
Shklovskii
, “
Compensation driven super-conductor-insulator transition
,”
Phys. Rev. B
79
,
134504
(
2009
).
234.
K.
Segawa
and
Y.
Ando
,
Phys. Rev. B
74
,
100508(R)
(
2006
).
235.
K. V.
Reich
,
M.
Schecter
, and
B. I.
Shklovskii
, “
Accumulation, inversion, and depletion layers in SrTiO3
.”
Phys. Rev. B
91
,
115303
(
2015
).
236.
Han
Fu
,
K. V.
Reich
, and
B. I.
Shklovskii
, “
Anomalous conductivity, Hall factor, magnetoresistance, and thermopower of accumulation layer in SrTiO3
,”
Phys. Rev. B
94
,
045310
(
2016
).
237.
M.
Sammon
,
Han
Fu
, and
B. I.
Shklovskii
, “
Anomalous thermodynamic properties of electron accumulation layer in SrTiO3
,”
Phys. Rev. B
96
,
155203
(
2017
).
238.
M.
Sammon
,
H.
Fu
, and
B. I.
Shklovskii
, “
Electron accumulation layer in super-strong magnetic field
,”
Fiz. Nizk. Temp.
43
,
283
(
2017
) [
Low Temp. Phys.
43, 232 (2017)].
239.
Han
Fu
,
K. V.
Reich
, and
B. I.
Shklovskii
, “
Collapse of electrons to a donor cluster in SrTiO3
,”
Phys. Rev. B
92
,
035204
(
2015
).
240.
I. M.
Lifshitz
,
S. A.
Gredeskul
, and
L. A.
Pastur
,
Introduction to the Theory of Disordered Systems
(
Wiley
,
New York
,
1988
).
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